Exercises: Properties of differentiable functions. Rolle's, mean value and Cauchy's theorems

Given the polynomial function $f(x) = x^2 - 6x + 8$, check whether the hypotheses of Rolle's Theorem hold on the interval $[2, 4]$. If they do, find the value of the point $c \in (2, 4)$ whose existence is guaranteed by the theorem.

Consider the function $f(x) = x^2 - kx + 10$. Determine the value of the parameter $k \in \mathbb{R}$ so that the function satisfies the condition $f(1) = f(5)$ and Rolle's Theorem can be applied on the interval $[1, 5]$. Once $k$ has been found, calculate the value of $c$ where the derivative is zero.

Consider the function $f(x) = x^2 - x$ on the interval $[0, 2]$. The figure shows its graph and the secant line through the points $A$ and $B$ of the graph with $x$-coordinates $0$ and $2$.

1. a) Check that the hypotheses of the Mean Value Theorem (Lagrange) hold on $[0, 2]$ and calculate the slope of the secant line $AB$.

1. b) Find the value of the point $c \in (0, 2)$ whose existence is guaranteed by the theorem.

1. c) Explain graphically how the tangent line to the graph at the point with $x$-coordinate $c$ is related to the secant line $AB$.

A marathon runner trains on a straight track. Their position (in metres) relative to the starting point is given by the function $s(t) = t^2 + 2t$, where $t$ is the time in seconds. Use the Mean Value Theorem to show that at some instant between $t=1$ and $t=4$ seconds, their instantaneous velocity was exactly equal to their average velocity on that interval. Calculate that instant.

Explica razonadamente por qué no se puede aplicar el Teorema de Rolle a la función $f(x) = 1 - |x-1|$ en el intervalo $[0, 2]$, a pesar de que $f(0) = f(2) = 0$. ¿Qué hipótesis falla en este caso?

Given the functions $f(x) = x^2 + 1$ and $g(x) = 2x - 3$, check whether the conditions of Cauchy's Theorem are satisfied on the interval $[1, 3]$. If so, find the value of $c \in (1, 3)$ such that:

It is known that a function $f(x)$ is continuous on the closed interval $[1, 5]$ and differentiable on the open interval $(1, 5)$. If the values of the function at the endpoints are $f(1) = 10$ and $f(5) = 30$, can we guarantee that there exists a point $c$ in the interval where the gradient of the tangent line is equal to $5$? Justify your answer using the Mean Value Theorem.

Given the function $f(x) = \sin(x)$, solve:

1. a) Verify that the hypotheses of Rolle's Theorem hold on the interval $[0, \pi]$.

1. b) Find the value of $c$ that satisfies the conclusion of the theorem.

1. c) Represent $c$ schematically on the graph of the function (you can copy the figure) or on a unit circle. If you use the graph, mark the point with abscissa $c$ and draw the tangent line at that point.

Mateo has to solve the following problem in an exam: «Check whether the function $f(x) = 1 - |x|$ satisfies the hypotheses of Rolle's Theorem on the interval $[-2, 2]$ and find the point $c$ predicted by the theorem». On his exam paper, Mateo writes:

«1. The function is continuous on $[-2, 2]$ because it is a composition of continuous functions.

1. The values at the endpoints coincide: $f(-2) = 1 - |-2| = -1$ and $f(2) = 1 - |2| = -1$.

2. Since the hypotheses are satisfied, Rolle's Theorem ensures that there exists $c \in (-2, 2)$ with $f'(c) = 0$. Given that the derivative is $f'(x) = 1$ for $x < 0$ and $f'(x) = -1$ for $x > 0$, the desired point is $c = 0$».

Identify the two serious conceptual errors made by Mateo (one when checking the hypotheses of the theorem and another when determining the value of the derivative at $c = 0$), explain in detail why they invalidate his conclusion, and determine whether or not there exists any point in said interval where the derivative vanishes.

Find with reasoning the coefficients $a, b \in \mathbb{R}$ of a quadratic function of the form $f(x) = ax^2 + bx$ that simultaneously satisfies the following conditions:

1. It satisfies all the hypotheses of Rolle's Theorem on the closed interval $[0, 6]$.

2. It attains a relative extremum at the interior point whose existence is guaranteed by the theorem, with that value being $f(c) = -18$.

Once the values of $a$ and $b$ are deduced, state precisely the hypotheses of Rolle's Theorem and formally verify that the resulting function satisfies them on $[0, 6]$, explicitly calculating the resulting point $c \in (0, 6)$.

Given the piecewise function:

Determine whether the hypotheses of the Mean Value Theorem are satisfied on the interval $[1, e+2]$ and, if so, find the value or values of $c \in (1, e+2)$ whose existence is guaranteed by the theorem.

An average speed camera on a Spanish motorway detects that a vehicle enters the Guadarrama tunnel (point $A$) at 14:00:00 and leaves it (point $B$) at 14:01:30. The distance between both control points is 3.5 km.

If the speed limit in the tunnel is $120 \text{ km/h}$, use the Lagrange Mean Value Theorem to justify whether the driver should receive a speeding fine, regardless of whether their speed at the entrance and exit was legal.

Consider the polynomial function $f(x) = x^3 + x^2 + kx$.

1. a) Determine the value of the parameter $k$ so that the function satisfies the hypotheses of Rolle's Theorem in the interval $[-2, 1]$.
2. b) For that value of $k$, calculate the points $c$ that satisfy the conclusion of the theorem.

Consider the function $f(x) = x \cdot e^{x^2-1}$.

1. a) Verify that $f(x)$ satisfies the hypotheses of the Mean Value Theorem on the interval $[0, 1]$.
2. b) Is there any point in that interval where the slope of the tangent line is equal to $1$? Justify your answer using the theorem.

Apply Cauchy's Theorem to the functions $f(x) = \sin(x)$ y $g(x) = \cos(x)$ on the interval $\left[ 0, \frac{\pi}{2} \right]$, that is, with $a = 0$ and $b = \frac{\pi}{2}$. Find the value of $c \in \left(0, \frac{\pi}{2}\right)$ that satisfies the equality:

Explain the geometric interpretation of this result in terms of the parametric curve $t \mapsto (g(t), f(t))$, with $t \in \left[0, \frac{\pi}{2}\right]$. The figure shows that curve, its endpoints $A = (g(a), f(a))$ and $B = (g(b), f(b))$, the secant line $AB$, and the increments $g(b)-g(a)$ and $f(b)-f(a)$.

Use Bolzano's Theorem and Rolle's Theorem to prove that the equation:

has exactly one real solution on the interval $[0, 1]$.

Consider the function $f(x) = \sqrt{x^2 + 9}$.

Find a point on the graph of the function where the tangent line is parallel to the secant line joining the points on the curve with x-coordinates $x=0$ and $x=4$. Which theorem guarantees that such a point exists? State its hypotheses and check that they are satisfied.

Given the function $f(x) = \frac{x^2 - 6x + 5}{x - 3}$, can Rolle's Theorem be applied on the interval $[1, 5]$ knowing that $f(1) = f(5) = 0$?

Justify your answer by analysing the continuity and differentiability of the function on that interval. If not, explain whether this contradicts the theorem.

The Dirección General de Tráfico (DGT) has an average speed camera installed on the AP-6 motorway to monitor a $15\text{ km}$ section where the legal maximum speed is set at $100\text{ km/h}$. A passenger car enters the section at $11:00:00$ recording an instantaneous entry speed of $90\text{ km/h}$, and exits the section at $11:08:15$ recording an instantaneous exit speed of $95\text{ km/h}$.

1. a) Assuming that the vehicle's position function $s(t)$ with respect to time is continuous and differentiable, calculate the average speed of the trip in $\text{km/h}$.
2. b) State Lagrange's Mean Value Theorem and use it to justify whether the DGT can legally fine the driver for speeding, explaining whether or not it is necessary to have an intermediate snapshot of the vehicle travelling above $100\text{ km/h}$.

Consider the function $f(x) = \sqrt{x}$ defined on the interval $[1, 9]$.

1. a) Before carrying out any calculation, make a reasoned estimation: knowing that the square root function is concave downward on its entire domain (that is, its first derivative is strictly decreasing and its tangent slopes keep decreasing), will the point $c \in (1, 9)$ at which the tangent line is parallel to the secant line joining the endpoints be located to the left of the arithmetic midpoint ($x = 5$), at the midpoint itself, or to its right? Justify your prediction qualitatively.
2. b) State Lagrange's Mean Value Theorem, verify that its hypotheses hold on $[1, 9]$, and calculate the exact value of $c$.
3. c) Check whether the numerical value obtained confirms your initial estimation and interpret the result geometrically.

Let the piecewise function be defined as:

$f(x) = \begin{cases} x^2 + ax + b & \text{if } x \le 1 \\ k \ln(x) & \text{if } x > 1 \end{cases}$

Calculate the values of the parameters $a, b, k \in \mathbb{R}$ so that the function $f(x)$ satisfies the hypotheses of Rolle's Theorem on the interval $[0, e]$. Once the parameters are found, determine the value of the point $c \in (0, e)$ whose existence is guaranteed by the theorem.

Prove, using both Bolzano's Theorem and Rolle's Theorem, that the equation:

has a unique real solution for any value of the constant $k \in \mathbb{R}$. Give a reasoned explanation of why the existence of more than one root would contradict the statement of Rolle's Theorem.

The GPS logging system of a freight train shows that it passed the 150 km mark of a railway line at 10:00:00 and the 162 km mark at 10:06:40. The speed limit on that line is $100 \text{ km/h}$.

1. a) Using the Mean Value Theorem (Lagrange), justify whether the train exceeded the speed limit on the line at any point.
2. b) If the function describing the train's position on that section is $s(t) = 150 + 2t - 0,03t^2$ (where $s$ is in km and $t$ in minutes since passing the first kilometre mark), find the exact instant at which the instantaneous velocity coincides with the average speed of the journey.

Given the functions $f(x) = e^x$ and $g(x) = x^2 + 1$, apply Cauchy's Theorem (Generalized Mean Value Theorem) on the interval $[0, 2]$.

Find the value of $c \in (0, 2)$, corresponding to point $P$ in the figure, that satisfies the conclusion of the theorem:

Consider a function $f(x)$ differentiable on all of $\mathbb{R}$. It is known that the equation $f'(x) = 0$ has exactly two real solutions, $x_1 = -2$ and $x_2 = 3$.

Prove that the function $f(x)$ cannot have more than three distinct real roots. Is it possible for the function to have only one real root? Justify your answer based on the geometric interpretations of differentiability theorems.

Use the Mean Value Theorem to prove the following inequality for any pair of real numbers $a$ and $b$ such that $0 < a < b$:

Hint: Apply the theorem to the function $f(x) = \ln(x)$ on the interval $[a, b]$.

Consider the function $f(x) = x^3 - 3x + m$.

1. a) Determine for which values of the parameter $m$ the function satisfies the hypotheses of Rolle's Theorem on an interval of the form $[0, a]$ with $a > 0$.
2. b) For $m = 0$, find the largest possible interval $[p, q]$ where the theorem can be applied and calculate the point $c$ whose existence is guaranteed.

Two runners, Alba and Bruno, start a 100-metre hurdles race at the same time and reach the finish line in a photo finish (same time $T$). Their position functions are $s_A(t)$ and $s_B(t)$, both continuous on $[0, T]$ and differentiable on $(0, T)$.

Prove, using Rolle's Theorem on the difference function $h(t) = s_A(t) - s_B(t)$, that there was at least one instant during the race when both runners had exactly the same instantaneous velocity.

It is desired to prove rigorously that the equation $2x^3 + 3x - 4 = 0$ has exactly one real solution in the open interval $(0, 2)$. To tackle this problem, two students propose different approaches:

• Strategy A (Bolzano and Monotonicity): Prove the existence of at least one root in $(0, 2)$ using Bolzano's Theorem and then prove that the associated function is strictly increasing on all of $\mathbb{R}$ by analysing the sign of $f'(x)$ to deduce uniqueness.

• Strategy B (Bolzano and Rolle's Theorem): Prove the existence of at least one root with Bolzano and then assume by contradiction that there exist two distinct roots $x_1 < x_2$ in $(0, 2)$, applying Rolle's Theorem on $[x_1, x_2]$ to reach a contradiction with the equation $f'(x) = 0$.

You are asked to:

1. a) Critically compare both strategies: argue why both are valid and explain which of the two is methodologically more direct when the first derivative has no real roots.
2. b) Choose one of the two strategies and write out the complete, rigorous proof of the existence and uniqueness of the root, explicitly verifying all required hypotheses of the theorems you use.

Given the functions $f(x) = \cos(x)$ and $g(x) = \operatorname{sen}(x)$ defined on the interval $\left[0, \frac{\pi}{2}\right]$:

1. a) Formally check whether all hypotheses of Cauchy's Mean Value Theorem (Extended Mean Value Theorem) are satisfied on said interval, rigorously justifying the condition on the non-vanishing of the denominator's derivative $g'(x)$ on the open interval.
2. b) Find the exact value of the point $c \in \left(0, \frac{\pi}{2}\right)$ whose existence the theorem guarantees.
3. c) Apply Lagrange's Mean Value Theorem separately to $f(x)$ and $g(x)$ on the same interval $\left[0, \frac{\pi}{2}\right]$, analytically determine the corresponding intermediate points $c_1$ and $c_2$, and prove that neither of them coincides with the value $c$ obtained via Cauchy's Theorem. Justify conceptually why Cauchy's Theorem is not equivalent to applying Lagrange independently to each function.